A Multiparameter Degeneracy in Microlensing Events with Extreme Finite Source Effects

Microlensing events can be subject to degeneracies, i.e., when one event can be described by two or more models equally well within given photometric measurements. There are two generic types of degeneracies: "accidental" degeneracies and "mathematical" degeneracies. In accidental degeneracies, the similarities between the models are not due to any fundamental underlying mathematical symmetry but rather just due to coincidence. Generally, such accidental degeneracies can be resolved with higher-precision photometric measurements or increased photometric monitoring. On the other hand, "mathematical" degeneracies exist because of some deeper underlying symmetries in the lens models. These symmetries typically appear in some extreme limits in one or more of the parameters that describe the model. For example, many of these degeneracies can be derived by expanding the lens equation ³ in some small parameter. By keeping only the lowest-order terms, the lens equation becomes degenerate with respect to that parameter or with the lens equation expanded in the same way with respect to another small parameter (see, e.g., Dominik 1999 for examples). Mathematical degeneracies are more nefarious than accidental ones because the underlying models for the magnification can become nearly perfectly degenerate in some extreme limits, and thus such degeneracies cannot be resolved even with exquisite data.

The nature of a degeneracy can be discrete or continuous. Discrete degeneracies occur when a finite number of models can be used to describe a given event. An infamous example is the s ↔ s⁻¹ degeneracy for low mass-ratio binary lenses, where s is the instantaneous projected semimajor axis of the binary a_⊥ in units θ_E s = (a_⊥/D_Lens)/θ_E (Griest & Safizadeh 1998; Dominik 1999; Yee et al. 2021). Here, θ_E is the angular Einstein ring radius, given by

$$\begin{eqnarray}&&{\theta }_{{\rm{E}}}=\sqrt{\kappa {M}_{\mathrm{lens}}{\pi }_{\mathrm{rel}}},\end{eqnarray} \tag{ 1 }$$

where M_lens is the lens mass, π_rel = au (1/D_Lens − 1/D_Source) is the lens-source relative parallax, and κ = 8.14 mas ${M}_{\odot }^{-1}$ is a constant.

Continuous degeneracies occur for events in which a range of parameters can be used to model an event. One example is the degeneracy between the impact parameter normalized to the Einstein ring radius u₀, the microlensing timescale t_E, and the fraction of flux attributed to the source relative to the combined source and blend flux f_S. Here, the microlensing timescale t_E is given by

$$\begin{eqnarray}&&{t}_{{\rm{E}}}=\displaystyle \frac{{\theta }_{{\rm{E}}}}{{\mu }_{\mathrm{rel}}}\end{eqnarray} \tag{ 2 }$$

where μ_rel is the relative lens-source proper motion, the impact parameter u₀ ≡ θ₀/θ_E is the angular distance of closest approach between the lens and source on the sky θ₀ in units of θ_E, and f_S = F_S/(F_S + F_B) where F_S and F_B are the source flux and any unresolved flux blended with the source flux, respectively. This degeneracy was first discussed in detail by Woźniak & Paczyński (1997), and operates in two regimes: when u₀ ≪ 1 and when u₀ ≫ 1. Another continuous degeneracy exists for bound planetary microlensing events between the lens mass ratio q = M_planet/M_host, the angular source size θ_* in units of θ_E

$$\begin{eqnarray}&&\rho \equiv \displaystyle \frac{{\theta }_{* }}{{\theta }_{{\rm{E}}}},\end{eqnarray} \tag{ 3 }$$

and μ_rel for a subset of perturbations from a bound planet, described by Gaudi & Gould (1997). Typically, the source star in a microlensing event can be well approximated as a point source. However, if there exists a significant second derivative of the magnification over the angular area covered by the source, the normalized angular size of the source star ρ must be included in the model, i.e., the event exhibits finite source effects (FSEs). Both of these continuous degeneracies are similar in nature to the one we report here, but differ in detail.

Here we explore a continuous degeneracy between multiple parameters that emerges for isolated lenses with ρ ≫ 1, which we refer to as extreme finite source effect (EFSE) events. Parts of this degeneracy have been identified in FFP candidate events (for which ρ is typically ≳1) reported in Mróz et al. (2018), Mróz & Udalski et al. (2019), Mróz et al. (2020a), and Mróz et al. (2020b). Specifically in Mróz et al. (2020a), when fitting a lensing model to their event, they noted a strong correlation between four parameters ρ, t_E, u₀, and f_S. As we elucidate here, the correlations identified by the above authors are due to a continuous, mathematical degeneracy between the parameters that describe EFSE microlensing events.

We note that this degeneracy has also been described in the context of self-lensing binary systems (Kruse & Agol 2014; Han 2016). Such systems can exhibit periodic brightening events when the remnant member of the binary passes in front of its main-sequence companion star. These brightening events are equivalent to EFSE events. This degeneracy is also qualitatively similar to the degeneracy between the impact parameter, radius ratio, and limb darkening in transit light curves (e.g., Morris et al. 2020). This is because the geometry and shapes of transit light curves are very similar to those of EFSE events, although transit light curves are obviously inverted related to EFSE events. Thus, strong analogies can be drawn between the two.

As we will show, in the EFSE regime and assuming no limb darkening, light curves can be characterized by three gross observables, but their complete models have four free parameters. Namely, these observables are the peak flux above the baseline of the event ${\rm{\Delta }}{F}_{\max }$ (which is approximately constant for events with ρ ≫ 1 when the center of the lens and source are separated by less than θ_*), the FWHM duration of the event t_FWHM, and the fraction of time spent in the wings/shoulders of an event f_ws. We note that there are two other observables, namely the baseline flux F_base = F_S + F_B, which is only constraining when multiband photometry is collected while the source is magnified, and t₀, which is time symmetric and is not a part of this degeneracy. Thus, in the absence of limb darkening and for single-band photometry, four free parameters characterize the flux as a function of time F(t) and there are only three observables, resulting in a degeneracy. These parameters can be analytically approximated to be pairwise degenerate. The source size ρ is degenerate with the source flux F_S in such a way that they can be varied in order to maintain the peak magnification of an event (${\rm{\Delta }}{F}_{\max }$). Similarly, the source star angular radius crossing time

$$\begin{eqnarray}&&{t}_{* }\equiv \displaystyle \frac{{\theta }_{* }}{{\mu }_{\mathrm{rel}}}\end{eqnarray} \tag{ 4 }$$

and the impact parameter scaled to the angular size of the source star b₀ = θ₀/θ_* are degenerate in that they can be varied to maintain the observed duration (t_FWHM) of an event.

When limb darkening is included however, the situation becomes more complex. A fourth observable becomes measurable, which is the flux ratio of the event at its peak to when the lens is centered on the limb of the source f_pl. For a linear limb-darkening profile with coefficient Γ, this introduces a fifth parameter for four observables, meaning a degeneracy remains. Furthermore, for nonzero Γ, ${\rm{\Delta }}{F}_{\max }$ now also depends on b₀, which results in the coupling between the four previous parameters into a larger five-parameter degeneracy through the variation in the flux during the event due to limb darkening.

In Section 2 we review the mathematical models, parameters, and observables of single-lens microlensing events in the ρ ≪ 1 regime. In Section 3, we repeat this process and describe the morphology and observables of EFSE events without limb darkening. We re-parameterize the canonical set of single-lens parameters to a new set of parameters that are more closely tied to the observables. We then derive the degeneracy for a single-lens event in the EFSE regime without considering limb darkening. We then add another observable and extend the degeneracy to include limb darkening in Section 4. We explore these degeneracies qualitatively and quantitatively for fixed limb darkening in Section 5, and explore the full degeneracy including limb darkening as a free parameter in Section 6. We then discuss physical constraints on the severity of these mathematical degeneracies in Section 7. Finally, we consider the implications of our findings and conclude in Section 8.

This paper is fairly long, and is written in a pedagogical style in order the provide the full context with which to understand the EFSE degeneracy. Experts in microlensing and/or readers that are not interested in the details of the degeneracy may want to first turn to Section 8 to read the summary and discussion of the main new results. Then, the reader can decide which sections of the paper they want to read (or skip). The different sections are written so that they can be read more or less independently.

Throughout this paper, we calculate the magnifications using either the Witt & Mao (1994) or the Lee et al. (2009) method as implemented in MulensModel (Poleski & Yee 2019).

For the majority of the single-lens microlensing events that have been observed to date, the size of the source star can be ignored, as its angular size is much smaller than the angular Einstein ring radius of the lens (i.e., ρ ≪ 1) and the source does not pass within ∼1 source radius of the lens. In this approximation, the point-source point-lens (PSPL) model magnification A_ps(t) is given by the standard formula (Paczynski 1986)

$$\begin{eqnarray}&&{A}_{\mathrm{ps}}(t)=\displaystyle \frac{u{\left(t\right)}^{2}+2}{u(t)\sqrt{u{\left(t\right)}^{2}+4}},\end{eqnarray} \tag{ 5 }$$

where $u{(t)}^{2}={u}_{0}^{2}+{\tau }_{{\rm{E}}}^{2}(t)$, and τ_E(t) ≡ (t − t₀)/t_E is the time from the peak of the event in units of the microlensing timescale. Note that, such events reach peak magnification at a time t₀ when u(t) is at its minimum angular separation u₀.

The magnification is not a direct observable; rather, one measures the flux as a function of time

$$\begin{eqnarray}&&F(t)={F}_{{\rm{S}}}A(t)+{F}_{{\rm{B}}}.\end{eqnarray} \tag{ 6 }$$

Note that F_B can include flux from the lens, any flux from companions to either or both the lens and source, as well as flux from unrelated stars that are blended in the point-spread function (PSF) of the source.

The point-source magnification in Equation (5) diverges as u → 0 and in this regime can be approximated by ${A}_{\mathrm{ps}}(t)\simeq {\left[u(t)\right]}^{-1}$. When u ≫ 1, the magnification can be approximated as A_ps(t) ≃ 1 + 2[u(t)]⁻⁴. As mentioned previously, Woźniak & Paczyński (1997) demonstrated that in these two limits, there is a continuous mathematical degeneracy between u₀, t_E, and f_S.

A PSPL microlensing event can be described by five parameters when FSEs are negligible: u₀, t₀, t_E, F_S, and F_B. To the extent that there are only four gross observables, namely t₀, the baseline flux F_base, the difference between the peak flux and the baseline flux ${\rm{\Delta }}{F}_{\max }$, and some measure of the characteristic timescale of the event (such as FWHM), it is clear that u₀, t_E, and f_S cannot be uniquely determined, and there is a continuous degeneracy in these parameters. This is the basic underlying cause of the degeneracy described by Woźniak & Paczyński (1997). In reality, when one is not deep in the limits noted above by Woźniak & Paczyński (1997), there are additional observables related to the detailed shape of the microlensing light curve that allow one to break this degeneracy with sufficiently good light-curve coverage and photometric precision.

The magnification of a finite source begins to deviate from Equation (5) when ρ ≳ 2u₀ (Liebes 1964; Gould & Gaucherel 1997). In this case, one must include the additional parameter ρ, as well as the limb-darkening profile of the source, which is commonly described as a linear limb-darkening profile with a coefficient Γ (Yoo et al. 2004). Thus, point-lens events can be described by seven parameters when FSEs are significant: u₀, t₀, t_E, F_S, F_B, ρ, and Γ. For completeness, we note that for stars in the bulge, and typical lenses with masses in the brown dwarf, stellar, or remnant regimes, ρ ≪ 1. Thus, FSEs only begin to manifest as deviations from the point-source magnification (Equation (5)) in high-magnification events when the source approaches within a few stellar radii of the lens, i.e., events for which u₀ ≲ ρ/2. These events are relatively rare. In this case, the majority of the light curve is well approximated by the point-source assumption, except for a deviation within a few source crossing times t_* of t₀. This deviation takes the form of a "rounding" of the peak of the microlensing event. Because the deviations due to FSEs are localized to a small time window near the peak, the parameters ρ and Γ do not participate in the single-lens degeneracy identified in Woźniak & Paczyński (1997).

We now derive the degeneracy for EFSE events for a source without limb darkening. As discussed previously, in the case of no limb darkening and ρ → ∞, there are two gross observables, namely ${\rm{\Delta }}{F}_{\max }$ (which is roughly constant during the event), and t_FWHM. Furthermore, these two observables are decoupled under these assumptions. We therefore first consider the flux degeneracy and duration degeneracy separately.

The maximum difference flux when Γ = 0 is simply

$$\begin{eqnarray}&&{\rm{\Delta }}{F}_{\max }=\displaystyle \frac{2{F}_{{\rm{S}}}}{{\rho }^{2}},\qquad ({\rm{\Gamma }}=0).\end{eqnarray} \tag{ 19 }$$

It is straightforward to verify that substituting the parameters

$$\begin{eqnarray}&&{F}_{{\rm{S}}}^{{\prime} }=\zeta {F}_{{\rm{S}}},\qquad \rho ^{\prime} ={\zeta }^{1/2}\rho \end{eqnarray} \tag{ 20 }$$

into Equation (19), we recover the same observable difference flux (e.g., ${\rm{\Delta }}{F}_{\max }^{{\prime} }={\rm{\Delta }}{F}_{\max }$). Therefore, the difference flux is constant under the transformations ${F}_{{\rm{S}}}^{{\prime} }\to \zeta {F}_{{\rm{S}}}$ and $\rho ^{\prime} \to {\zeta }^{1/2}\rho$ for any arbitrary positive constant ζ. Thus in the limit of ρ → ∞ , there is a perfect degeneracy between F_S and ρ such that

$$\begin{eqnarray}&&{F}_{{\rm{S}}}^{{\prime} }={F}_{{\rm{S}}}{\left(\displaystyle \frac{\rho ^{\prime} }{\rho }\right)}^{2}.\end{eqnarray} \tag{ 21 }$$

It is also trivial to show that by dividing both sides of Equation (21) by F_base that the relationship holds for two blending parameters f_S and ${f}_{{\rm{S}}}^{{\prime} }$ such that

$$\begin{eqnarray}&&{f}_{{\rm{S}}}^{{\prime} }={f}_{{\rm{S}}}{\left(\displaystyle \frac{\rho ^{\prime} }{\rho }\right)}^{2}.\end{eqnarray} \tag{ 22 }$$

It can be more intuitive to express the degeneracy in terms of f_S rather than F_S, as this is a dimensionless parameter.

Now consider the duration of the event as parameterized by the source half-chord crossing time, which is related to the model parameters by t_c = β t_* (Equation (15)). It is trivial to verify that substituting the following parameters

$$\begin{eqnarray}&&\beta ^{\prime} =\xi \beta ,\qquad {t}_{* }^{{\prime} }={\xi }^{-1}{t}_{* }\end{eqnarray} \tag{ 23 }$$

into Equation (15) will result in an equal source chord crossing time (${t}_{{\rm{c}}}^{{\prime} }={t}_{{\rm{c}}}$) for any arbitrary positive constant ξ satisfying 0 ≤ ξ β ≤ 1. Thus, in the limit of ρ → ∞ , there is a perfect degeneracy between b₀ and t_* such that

$$\begin{eqnarray}&&{t}_{* }^{{\prime} }={t}_{* }\displaystyle \frac{\beta }{\beta ^{\prime} }={t}_{* }\displaystyle \frac{\sqrt{1-{b}_{0}^{2}}}{\sqrt{1-{b}_{0}{{\prime} }^{2}}}.\end{eqnarray} \tag{ 24 }$$

Because the duration of the event is decoupled from the flux during the event, ξ does not need to be equal to ζ (and, in general, will not be).

The above mathematical degeneracies are only strictly valid in the limit ρ → ∞ , or equivalently in the limit that t_E/t_* → 0. As discussed previously, for values of ρ that are large but finite, EFSE events deviate from the strict top-hat morphology. In particular and as illustrated in Figures 1 and 2, events with finite ρ exhibit wings and shoulders near the source limb crossing point of the event. The duration of these features relative to the half-chord crossing time is equal to ${f}_{\mathrm{ws}}={({\beta }^{2}\rho )}^{-1}$ (see Equations (17) and (33), and the surrounding discussion). Effectively, f_ws provides another observable parameter when ρ is large but finite.

The fact that the duration of the wings and shoulders relative to t_c depends on both β and ρ has two important implications. First, the conclusion that F_S and ρ have mathematically equivalent effects on the peak flux and morphology of EFSE events provided that they satisfy Equation (21) is not strictly true. While varying F_S does not change the morphology of EFSE events, varying ρ does as f_ws is a function of ρ, but not a function of F_S. This is illustrated in Figure 1, which shows how as ρ increases, the duration of the wings and shoulders relative to the total duration of EFSE events decreases, and the morphology of the light curves becomes increasingly well approximated by a strict top hat.

Second, because f_ws depends on β and thus the impact parameter b₀, the morphology of events with large but finite ρ is not strictly independent of b₀. In particular, events with larger impact parameter (smaller β) exhibit more pronounced wings and shoulders. Thus, for EFSE events with large but finite values of ρ, the morphology of the event depends on b₀, and thus the flux during the event is coupled to the duration of the event. In the specific case when Γ = 0 and ρ is large but finite, there are four parameters (F_S, ρ, β, t_*) and three observables (${\rm{\Delta }}{F}_{\max },{t}_{{\rm{c}}},{f}_{\mathrm{ws}}$). Given the definitions of the observables, and assuming that the morphology of the wings and shoulders is directly proportional to their fractional duration f_ws, it is straightforward to show that there is a continuous mathematical one-parameter degeneracy. Specifically, by substituting the following parameters

$$\begin{eqnarray}&&{F}_{{\rm{S}}}^{{\prime} }=\eta {F}_{{\rm{S}}},\,\rho ^{\prime} ={\eta }^{1/2}\rho ,\,\beta ^{\prime} ={\eta }^{-1/4}\beta ,\,{t}_{* }^{{\prime} }={\eta }^{1/4}{t}_{* }\end{eqnarray} \tag{ 25 }$$

into Equations (15), (19), and (18), we recover the same values for the observables (i.e., ${\rm{\Delta }}{F}_{\max }^{{\prime} }={\rm{\Delta }}{F}_{\max },{t}_{{\rm{c}}}^{{\prime} }={t}_{{\rm{c}}},{f}_{\mathrm{ws}}^{{\prime} }={f}_{\mathrm{ws}}$). Here η is any arbitrary positive constant satisfying 0 ≤ η^−1/4 β ≤ 1.

We illustrate the mathematical severity of the degeneracy between these four variables in Figure 3. We use fiducial values of b₀ = 0.0, μ_rel = 6.5 mas yr⁻¹, and f_S ≃ 0.13 but use ρ = 3.6 in the top panel and ρ = 11.4 in the bottom panel. We then scale the other parameters by using nine uniformly spaced values of b₀ from 0.0 to 0.8, transforming these to values of β, and calculating η using Equation (25). In the upper subpanels, we show the resulting light curves, and in the lower subpanels, we show their relative residuals compared to the b₀ = 0.0 event, which is the lightest gray and backmost light curve. In both the top and bottom panels, the shading of the light curves and their residuals goes from lightest to darkest for increasing values of b₀. We see that the light curves are nearly perfectly degenerate, and in particular the ρ = 11.4 case are noticeably more degenerate than those for the ρ = 3.6 case. The prominent departures in the ρ = 3.6 case are due to the difference in the wing and shoulder shapes of the events due to detailed differences in the light curve caused by increasing ρ or β (centered on τ_c = ±1). Furthermore, there is a "trough" between the two peaks in the residuals that results from the slight decrease in flux as the impact parameter approaches the limb of the source star. However, these residuals decrease significantly in the ρ = 11.6 case. From this we conclude that for ρ ≳ 10, the morphology of these wings and shoulders does scale approximately with f_ws, and the mathematical degeneracy in Equation (25) is perfectly realized in the limit ρ → ∞ .

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Although the wings and shoulders of EFSE events due to finite values of ρ indeed provide a formal constraint on ρ and β, the magnitude of the differences in the light curves for impact parameters in the range b₀ = 0 − 0.8 is extremely small, as can be seen in the lower panels of Figure 3. For the ρ = 3.6 case, the full range of the deviation of the light curves relative to the fiducial (b₀ = 0) case is ≲1 × 10⁻³, whereas for the ρ = 11.4 case, it is ≲5 × 10⁻⁵. In both cases, these deviations are a factor of ∼40 times smaller than the magnitude of the event itself, and thus if the event is detected with a signal-to-noise ratio (S/N) = X, the deviations will only be detectable at S/N ≲ X/40. Thus, even for events that are detected at very high S/N, the deviations will be essentially undetectable, and the mathematical degeneracy in Equation (25) will hold.

However, we note that Figure 3 may make the degeneracy appear more pernicious than is likely to be realized in practice. In particular, and as we discuss in more detail in Section 7, when determining how deleterious these mathematical degeneracies will be for actual detected events, one must also consider the expected prior distributions for the underlying parameters. For example, for microlensing events with giant sources, the fraction of the baseline flux due to the source is known to be bimodal (Mróz et al. 2020a) such that the source flux is either very close to the baseline flux (i.e., the blend flux is very small) or the source flux is much smaller than the baseline flux (i.e., the majority of the baseline flux is due to the blend). This is simply because the luminosity function of stars in the bulge has a local minimum between giants and the main-sequence turn-off. As a result, EFSE events with giant sources will generally have F_S ≃ F_base as strongly blended events will be difficult or impossible to detect. This is important because events with F_S ≃ F_base and β ≃ 1 (b₀ ≃ 0) are not strongly subject to the degeneracy in Equation (25), since the source flux is bounded such that F_S/F_base ≲ 1 and the impact parameter is bounded such that b₀ ≥ 0 (β ≤ 1), implying that there is a narrow range of η that can satisfy Equation (25), which shrinks to nearly zero as β → 1 and F_S/F_base → 1. Furthermore, events with larger values of b₀ (smaller values of β) are less likely to be detected because they have a shorter duration and smaller peak difference flux (all else being equal). Finally, even when the degeneracy is realized, the posterior distribution of, e.g., F_S, will be narrower than the full allowed range. This is because events occur (although are not necessarily detected) with uniform values of b₀, which leads to a distribution of η that is not uniform, and in particular is weighted toward smaller values of η. This also implies that the distributions of F_S, ρ, β, and t_* are not uniform, and in particular are weighted toward smaller, smaller, larger, and smaller values, respectively.

Another example of prior information that is important to note is that we placed no upper limit on the value of f_S, therefore allowing F_S > F_base. This is commonly known as negative blending, as F_B must be less than zero for F_S = F_base − F_B > F_base (e.g., Smith et al. 2007). While seemingly unphysical, in crowded fields such as those toward that Galactic bulge that are typically monitored by microlensing surveys, it is possible to have some negative blending if the source happens to be located in a local minimum in the background that is typically dominated by an inhomogeneous "sea" of partially resolved faint stars. However, very large negative blending, i.e., F_S ≫ F_base or f_s significantly greater than unity, is essentially never realized in nature. Therefore, we will place an upper limit on the value of f_S in Section 5 and beyond.

We now consider the degeneracies that exist for a limb-darkened source (Γ ≠ 0), assuming the limb-darkening parameter Γ is known a priori and thus is not a free parameter. We preface this discussion by noting that for a limb-darkened source, the peak flux and shape of the EFSE event depend not only on F_S and ρ, but also on the impact parameter b₀. This is because the peak flux observed during the event now depends on the location of the center of the lens with respect to the center of the source, such that larger lens-source separations result in smaller peak fluxes (see Equation (30)). On the other hand, as with the Γ = 0 case, the observed duration t_c is related to t_* and b₀. Thus the duration and magnitude of the deviation are no longer decoupled. Nevertheless, as we will show, there is an approximate degeneracy in the case of fixed limb darkening that becomes a perfect mathematical degeneracy as Γ → 0 and another perfect mathematical degeneracy as Γ → 1. As we will show, this approximate degeneracy is actually quite severe.

We first recall that there are four primary observables ⁴ (${\rm{\Delta }}{F}_{\max }$, t_c, f_ws, and f_pl), and for fixed limb darkening with known Γ, there are four free parameters (b₀, t_*, F_S, and ρ). Thus, given that there are an equal number of observables as free parameters, we might anticipate that there would not be a degeneracy. From Equation (33) and assuming fixed Γ ≠ 0, a measurement of f_pl yields a constraint on β and thus b₀. A measurement of t_c and a constraint on β thus yields a constraint on t_*. A measurement of f_ws and a constraint on β also yields a constraint on ρ. Finally, a measurement of ${\rm{\Delta }}{F}_{\max }$, combined with a constraint on ρ, yields a constraint on F_S. Thus there is no mathematical degeneracy.

However, the lack of mathematical degeneracy rests on the fact that f_pl depends on b₀. Therefore, we next explore how the observable f_pl depends on b₀ for various values of Γ in order to provide a qualitative understanding of how well β can be constrained with a measurement of f_pl of a given precision. Figure 4 shows f_pl as a function of b₀ for various values of Γ = 0.0, 0.1, 0.2,...,1.0, where Γ = 0.0 is the darkest line and Γ = 1.0 the lightest. There are several points to note. First, when Γ = 0, f_pl = 0 and thus is independent of b₀, as noted previously. Second, for nonzero Γ, f_pl is a weak function of b₀ over a relatively broad range of b₀. Furthermore, the range of b₀ for which the difference in f_pl is smaller than some fixed value is larger for larger Γ. Thus the shape of the light curves become increasingly degenerate as Γ increases.

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For Γ = 1, the light curves are completely self-similar. This is due to the fact that for Γ = 1, f_pl = 1, and therefore ${\rm{\Delta }}{F}_{\max }=3{F}_{{\rm{S}}}\beta /{\rho }^{2}$, and ${\rm{\Delta }}F(t)={\rm{\Delta }}{F}_{\max }{T}_{c}(t)H(1-| {\tau }_{c}| )$ (see Section 4). Thus for Γ = 1, there are only three observables (${\rm{\Delta }}{F}_{\max }$, t_c, f_ws), and four parameters (F_S, ρ, β, t_*). All four parameters are therefore degenerate with each other, such that the observables are unchanged (i.e., ${\rm{\Delta }}{F}_{\max }^{{\prime} }={\rm{\Delta }}{F}_{\max }$, ${t}_{{\rm{c}}}^{{\prime} }={t}_{{\rm{c}}}$, and ${f}_{\mathrm{ws}}^{{\prime} }={f}_{\mathrm{ws}}$) under the transformation

$$\begin{eqnarray}\begin{array}{rcl}{F}_{{\rm{S}}}^{{\prime} } & = & \eta {F}_{{\rm{S}}},\,\beta ^{\prime} ={\eta }^{-1/5}\beta ,\\ \rho ^{\prime} & = & {\eta }^{2/5}\rho ,\,{t}_{* }^{{\prime} }={\eta }^{1/5}{t}_{* },\end{array}\end{eqnarray} \tag{ 35 }$$

for any arbitrary positive value of η that satisfies 0 ≤ η^−1/5 β ≤ 1.

Thus for fixed Γ = 0, there is a continuous one-parameter mathematical degeneracy between b₀, t_*, F_S, and ρ (Equation (25)), whereas for Γ = 1, there is also a continuous one-parameter mathematical degeneracy between these parameters, albeit with a different scaling (Equation (35)). Therefore, for intermediate values of Γ, we expect an approximate degeneracy, such that for smaller values of Γ, the scalings should more closely approximate Equations (25), whereas for larger values of Γ, the scalings should more closely approximate those of Equations (35). Indeed, this is what we find via numerical investigations (see Section 5 and the Appendix).

Finally, we consider the full degeneracy that exists for a limb-darkened source (Γ ≠ 0) when the limb darkening as parameterized by Γ is either completely unconstrained, or has some finite uncertainty. In the case of ρ → ∞ (and thus f_ws = 0), we still have three primary observables (${\rm{\Delta }}{F}_{\max }$, t_c, and f_pl), but we now have five free parameters (F_S, ρ, β, t_*, and Γ).

We proceed in the same manner as the previous sections. Consider an event with fiducial values of β, t_*, F_S, ρ, and Γ. Given the definitions of the observables, it is straightforward to show that there is a continuous mathematical one-parameter degeneracy between β, t_*, Γ, and F_S ρ⁻² such that

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}^{{\prime} } & = & \eta {\rm{\Gamma }},\,\beta ^{\prime} =\gamma \beta ,{t}_{* }^{{\prime} }={\gamma }^{-1}{t}_{* },\\ \left(\displaystyle \frac{{F}_{{\rm{S}}}}{{\rho }^{2}}\right)^{\prime} & = & \displaystyle \frac{1}{\eta \gamma }\left(\displaystyle \frac{{F}_{{\rm{S}}}}{{\rho }^{2}}\right),\end{array}\end{eqnarray} \tag{ 36 }$$

where we have defined ⁵

$$\begin{eqnarray}&&\gamma \equiv \displaystyle \frac{1-\eta {\rm{\Gamma }}}{\eta (1-{\rm{\Gamma }})},\end{eqnarray} \tag{ 37 }$$

and η is an arbitrary positive constant such that ηΓ ≤ 1 and γ β ≤ 1. Thus, F_S and ρ only participate in the degeneracy through the combination of F_S/ρ², and are degenerate with each other such that ${\rm{\Delta }}{F}_{\max }$ is constant under the transformations in Equation (20) for fixed ${\left(\eta \gamma \right)}^{-1}$.

We next consider the case when ρ ≫ 1 but finite, in which f_ws is nonzero. In this case, we still have five free parameters (F_S, ρ, β, t_*, and Γ), but there are now four primary observables (t_c, f_pl, f_ws, and ${\rm{\Delta }}{F}_{\max }$). We again consider an event with fiducial values of the parameters. Given the definitions of the observables, it is straightforward to show that there is a continuous mathematical degeneracy between F_S, ρ, β, t_*, and Γ such that

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}^{{\prime} } & = & \eta {\rm{\Gamma }},\,\,\beta ^{\prime} =\gamma \beta ,{t}_{* }^{{\prime} }={\gamma }^{-1}{t}_{* },\\ \rho ^{\prime} & = & {\gamma }^{-2}\rho ,{F}_{{\rm{S}}}^{{\prime} }={\eta }^{-1}{\gamma }^{-5}{F}_{{\rm{S}}},\end{array}\end{eqnarray} \tag{ 38 }$$

and η is an arbitrary positive constant such that ηΓ ≤ 1 and γ β ≤ 1. In this case, F_S and ρ become coupled to the larger degeneracy through the observables f_ws and ${\rm{\Delta }}{F}_{\max }$.

Analogous to Figure 3, we demonstrate the severity of this degeneracy in Figure 5. We use fiducial values of b₀ = 0.0, μ_rel = 6.5 mas yr⁻¹, f_s ≃ 0.13, and Γ = 0.2 but use ρ = 3.6 in the top panel and ρ = 11.4 in the bottom panel. Rather than scale the other parameters based on b₀, we use nine values that uniformly sample Γ from 0.2 to 0.4. The upper subpanels show the resulting light curve, whereas the fractional residuals compared to the Γ = 0.2 light curves are shown in the lower subpanels. In both the top and bottom panels, the color of the light curves and their residuals goes from lightest to darkest for increasing values of Γ. Similar again to Figure 3, the prominent deviations in the top panel are due to the changing values of ρ and β impacting the observable f_ws. But for the ρ = 11.4 case, we see the magnitude of these deviations decreases significantly. We also note that the "trough" between the peaks in the ρ = 3.6 case between τ_c = − 1 and 1 has a larger amplitude than in Figure 3. This is due to effects of higher order that affect the shape of the light curve induced by the limb-darkening profile at larger impact parameters. While light curves arising from different chords for a linear limb-darkening profile have similar shapes, they are not identical except when Γ = 0 or Γ = 1 (see Figure 4). We also note that these scalings require extreme values of the degenerate parameters. For instance in the ρ = 11.4 case, increasing Γ from 0.2 to 0.4 requires ρ ≈ 80 and f_S ≈ 10, the latter of which is unphysical.

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We reiterate that so far we have explored these degeneracies as purely mathematical degeneracies. In reality, b₀, t₀, F_S, ρ, and Γ are not free to take on an arbitrary range of values, alone and particularly when constrained by the values of the other parameters. For example, since the operand of ${ \mathcal S }(b)$ can only vary between the values of b = [0, 1], ${ \mathcal S }(b)$ can only take on the values between ${ \mathcal S }(0)=1+{\rm{\Gamma }}/2$ and ${ \mathcal S }(1)=1-{\rm{\Gamma }}$. Therefore, even by setting b₀ = 0, it is not possible to reproduce the maximum magnification of $A=1+\tfrac{2{F}_{{\rm{S}}}}{{\rho }^{2}}(1-{\rm{\Gamma }}/2)$ of such a light curve with another light curve with finite Γ by changing b₀. This is because the maximum magnification of a light curve with b₀ = 0 is indeed $A=1+\tfrac{2{F}_{{\rm{S}}}}{{\rho }^{2}}(1-{\rm{\Gamma }}/2)$, Γ has a maximum value of unity, and all light curves with b₀ > 0 have smaller maximum flux differences as the trajectory of the lens passes over only more limb-darkened portions of the source star. However, it would be possible by increasing f_S or decreasing ρ, although doing so may lead to either nonphysical or physically unlikely values of the other parameters.

When exploring the mathematical degeneracies in Sections 5 and 6, we adopt the following constraints: 0 ≤ Γ ≤ 1, b₀ ≥ 0 (β ≤ 1), and F_S > 0 (or f_s > 0). However, we allow the blend flux to be negative; specifically, we constrain f_s ≤ 3. As discussed in Section 7, somewhat negative blend fluxes are possible, and indeed have been observed in some events.

We further discuss the extent to which the mathematical degeneracies analyzed analytically in the previous sections and explored numerically in Sections 5 and 6 may manifest themselves observationally in Section 7.

In summary, there are seven model parameters that describe EFSE events when a linear limb-darkening profile is assumed: (1) the time at which the lens is minimally separated from the center of the source at the midpoint of the event t₀, (2) the source radius crossing time t_*, (3) the angular impact parameter of the lens relative to the center of the source in units of the angular radius of the source b₀, which we will generally parameterize as β for convenience, ⁶ (4) the baseline flux F_base, (5) the flux of the source F_S, which is related to the blend flux F_B through F_S = F_base − F_B, (6) the dimensionless radius of the star in units of the Einstein ring radius ρ, and (7) the dimensionless limb-darkening coefficient for a linear profile Γ. We emphasize again that, in the extreme case we are considering here, the light curve is better parameterized by b₀ and t_*, rather than by u₀ and t_E (Agol 2003).

Although the seven model parameters t₀, t_*, β, F_base, F_S, ρ, and Γ are all technically free parameters, only five (t_*, β, F_S, ρ, and Γ) participate in the degeneracies explored mathematically in Section 4.2. For well-sampled events, the parameter t₀ is not covariant with the other observables, because the derivative of the flux with respect to t₀ is antisymmetric about t₀, whereas the derivatives of the flux with respect to the other parameters are symmetric about t₀. Thus t₀ is (essentially) directly constrained by observations. As discussed above, F_base ≡ F_S + F_B is also essentially a direct (and typically precisely measured) observable based on observations outside of the microlensing event.

On the other hand, there are five gross observables for EFSE events for ρ → ∞ , namely: (1) the baseline flux F_base, (2) the peak change in flux ${\rm{\Delta }}{F}_{\max }$, (3) the FWHM of the event, which is roughly equal to the half-chord crossing time, t_c ≃ t_FWHM/2, (4) the fractional peak-to-limb flux difference f_pl, and (5) the midpoint t₀ of the (time-symmetric) event. Since F_base and t₀ are both parameters and direct observables, they do not participate in the degeneracy, and thus we only have to consider three observables: ${\rm{\Delta }}{F}_{\max }$, f_pl, and t_c. When ρ ≫ 1 but finite, we have a fourth observable, which is the duration of the event that exhibits wings/shoulders during the event f_ws.

In summary, for ρ ≫ 1 but finite, fixed limb darkening, and arbitrary Γ, there are an equal number of parameters and observables and thus there is no formal mathematical degeneracy. For the special cases of Γ = 0 and Γ = 1, there is one more free parameter than observables, and thus there is a one-parameter degeneracy in each case. However, the scalings of the parameters required to keep the observables fixed are different in the two cases (Equations (25) and (35)). In the case of free limb darkening, there is again one more parameter than observable, and thus a one-parameter degeneracy.

In order to investigate the nature of the degeneracies that we have explored mathematically in the previous section, we will qualitatively and quantitatively explore two fiducial events. For each, we will present a suite of light curves that highlight the various steps we have used to approach our explanation of the degeneracies. We also perform a quantitative investigation of the severity of the degeneracy with similar groupings of relevant parameters incorporated in the degeneracy.

5.1.1. Fiducial Events

For all of the following considerations, we will be comparing light curves to two "fiducial" light-curve models. For the first of these models, we select parameter values such that the physical limits of the degeneracy will fundamentally limit its mathematical extent and that are more akin to recent FFP candidates (e.g., Mróz et al. 2020a). For the second, we choose parameters that will allow us to explore the degeneracy more thoroughly within its physical bounds. We will refer to these fiducial models as "Event 1" and "Event 2" throughout the remainder of this paper. All Figures pertaining to Event 1 use green accents, while Figures pertaining to Event 2 use red accents. As discussed above, we note that in some cases, the degenerate light curves we find will require parameter values that are not physical. Such situations can therefore be ruled out, and a physical degeneracy (as opposed to a mathematical degeneracy) will not exist. Similarly, in other cases, some of the degeneracies will require parameter values that are physically unlikely. Such situations can therefore be considered implausible, and thus although both a physical and mathematical degeneracy exist, it is unlikely that this degeneracy will ever be manifested in actual microlensing surveys. Such situations can be dealt with by applying appropriate prior information. To allow the degeneracy to demonstrate a subset of these unlikely or impossible regions, we use a prior that f_S < 3.0 for all investigations. Note that this prior was not applied in Figures 3 and 5.

The parameter values we adopt for Event 1 are f_S = 1.00, f_B = F_B/F_base = 0.00, Γ = 0.4, b₀ = 0.0, π_rel = 0.125 mas, M = 0.11 M_⊕, ρ = 10, R_* = 10 R_⊙, θ_* = 5.8 μ as, t_* = 7.8 days, and μ_rel = 6.5 mas yr⁻¹. For Event 2, all of the parameters are identical except f_S = 0.20, f_B = 0.80, M = 0.55 M_⊕, and ρ = 4.47. These fiducial light curves are shown as the backmost light curve (green/red) in the third-from-the-top group (Γ = 0.4) of Figure 6. In the following figures, we specify which parameters vary from this fiducial set.

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5.1.2. Explanation of Figure Layouts

In each of the following subsections, two figures depict qualitative and quantitative representations of different aspects of the degeneracy for Events 1 and 2. The first of these of these two Figures contains two panels that show a qualitative comparison of sets of light curves with fixed limb darkening. The majority of the light curves within a set lie on top of each other, making distinguishing them difficult. The color order is constant, so when a list of values is provided, the respective line-coloring is green/red, black, dark gray, light gray, and white. Figures 6, 8, and 10 show six sets of light curves, each with fixed linear limb-darkening parameters of Γ = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0. We will relax the assumption of fixed limb darkening in Section 6. The first set of light curves in each Figure also includes five fractional uncertainties in the flux, which have values of 1%, 0.667%, 0.5%, 0.333%, and 0.1% from left to right. These uncertainties span the range of typical photometric uncertainties of ground-based FFP candidates (e.g., Figure 1 of both Mróz et al. 2020a, 2020b) to the smallest expected uncertainties for space-based discoveries (e.g., by the Roman Space Telescope in Figure 2 of Johnson et al. 2020). This visual comparison of the light curves demonstrates how similar they are, but leads to little understanding how severe the degeneracy is quantitatively.

We therefore proceed to numerically evaluate the severity of these degeneracies. We do so using the Δχ² statistic, which simply evaluates the difference in χ² between a proposed model and the fiducial model above for the simulated data. In such a scheme, broad swaths of parameter space may have ${\rm{\Delta }}{\chi }^{2}={\chi }_{\mathrm{trial}}^{2}-{\chi }_{\mathrm{fiducial}}^{2}$ that is significantly less than ∼9, or 3σ, leading to poor constraints on the subset of degenerate parameters for those events. The second figure in each subsection shows a map of Δχ² values for a much finer grid of the relevant parameters for each both fiducial cases. These are shown in Figures 7, 9, and 11, where we include gray-scaled maps of the Δχ² for a proposed model compared to the fiducial model under varying treatments for a labeled, fixed value of the limb-darkening parameter. Across all of these Figures, we adopt a uniform σ = 0.5% photometric uncertainty and a 15 min observation cadence in order to evaluate Δχ². We do not introduce any noise in the photometric data points, such that each lies perfectly on the model of the event. This is appropriate as it represents the severity of the degeneracies when averaged over a large ensemble of light-curve realizations. We parameterize our calculations in terms of the blending parameter f_S such that we can ignore units of flux. We limit f_S ≤ 3.0, but note that the negative blend fluxes implied by source fluxes at the upper end of this range are unlikely to be physical. Finally, we note that since we are adopting a constant fractional uncertainty, Δχ² is proportional to the rms fractional difference between the trial light curve and the fiducial light curve.

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The top panels of Figures 7, 9, and 11 have four subpanels, each corresponding to a fixed value of Γ = 0.0, 0.2, 0.4, or 0.6 as labeled in the subpanel. In these Figures, we sample t_* = θ_*/μ_rel rather than μ_rel, but as we hold θ_* constant, these are effectively interchangeable. In each subpanel, we indicate the appropriate fiducial model parameters with a green/red circle. We plot the trace of the minimum Δχ² as a solid green/red line. We also plot the appropriate analytic prediction (Equations (21) or (24)) for the degeneracies as a dashed green/red line. Note that all axes in these Figures are logarithmically scaled except for b₀.

In the bottom panels of Figures 7, 9, and 11, we include a plot of the marginalized minimum Δχ² as a function of the ordinate f_S or b₀ (the solid green/red line in the upper panels). In these bottom panels, each value of Γ = 0.0, 0.2, 0.4 and 0.6 uses a line style of solid, dashed, dotted–dashed, and dotted, respectively.

We begin by examining the degeneracy that results from constraining ${\rm{\Delta }}{F}_{\max }$, which depends on both ρ and F_S. In Figure 6 we fix the impact parameter b₀ = 0 for each set of light curves. For each Γ, five values of f_S = 1.0, 0.8, 0.6, 0.4, and 0.2 are plotted from the backmost to the frontmost light curve in the left panel for Event 1. The order of these f_S values is reversed in the right panel for Event 2. For each, we find the value of ρ that minimizes Δχ² with respect to the fiducial model. Note that to keep the timescale constant, θ_* and μ_rel are held constant, and we are only varying ρ by changing θ_E. Also note that the backmost (green/red) light curves in the Γ = 0.4 sets are the fiducial light curves for Event 1 and Event 2. In the Γ = 0.0 set, we see a slight rounding during the event as f_S decreases in the left panel, and the opposite on the right. This is caused by the compensating increase (decrease) in ρ through θ_E, which causes a departure from (to) the EFSE regime. However, as Γ increases to 0.6, the shape induced by the presence of limb darkening decreases this discrepancy near the peak of the event. For all values of Γ, as ρ decreases, the wings and shoulders of the event become more prominent as f_ws increases (Equation (18)). This is again due to the increase in the size of the Einstein ring, which begins to magnify the source at earlier times during the event. In both panels, we see that even just by varying ρ and f_S that the ${\rm{\Delta }}{F}_{\max }$ degeneracy is quite severe, with fractional differences on the order of 0.1%.

Next we perform the quantitative investigation of the ρ − F_S degeneracy. In Figure 7 we fix all event parameters to those of the fiducial models and find the Δχ² for each (ρ, f_S) pair in the grids mapped in the upper panels. We perform no minimization in computing the Δχ² of these grids. In both of the upper panels, we see a region of (ρ, f_S) pairs that follows the analytic prediction between ρ and f_S (Equation (21), dashed green/red line). We see that the analytic prediction of the minimum Δχ² nearly follows the numerical values except for a departure at small ρ where the events leave the EFSE regime (e.g., ρ becomes comparable to unity). In the upper-right panel (Event 2), the range of (ρ, f_S) pairs that have Δχ² < 9 increases significantly as Event 2 has values that better explore the physical extent of the degeneracy. In the bottom panels, we see that for both Events 1 and 2, the numerical minimum traces of Δχ² projected onto f_S have large ranges with values less than 9 (3σ). The numerical traces of Δχ² for Event 2 are lower than those for Event 1, especially for small f_S where Event 1 would require values of ρ that depart from the EFSE regime. However, both Events 1 and 2 have definite minima at the fiducial values of f_S. Also note that as Γ increases, we see a slight increase in the range of f_S values that have Δχ² < 9 as the shape induced in the light curve by the limb darkening of the source better matches the topology of the peak of the light curve (see Figure 6).

For both Events 1 and 2, we see that by only altering ρ and F_S, the degeneracy is strong but resolvable. These two parameters can alter ${\rm{\Delta }}{F}_{\max }$ of an event, but the detailed shape of the light curve will lead to a potential resolution in this aspect of the degeneracy if the photometric precision is sufficiently high.

Next we investigate the piece of the degeneracy that results from constraining t_FWHM, which is the trade-off between t_* and b₀. However, in this section we are only matching the timescale of the event; in particular, we are not modifying the ${\rm{\Delta }}{F}_{\max }$ of the event through changing either ρ or F_S. In both panels of Figure 8, we plot the six sets of light curves for each value of Γ. For each Γ, five values of b₀ = 0.0, 0.2, 0.4, 0.6, and 0.8 are plotted from the backmost to the frontmost light curve. We then vary t_* to match t_FWHM without accounting for the change in flux. Note that for increasing impact parameters, ${\rm{\Delta }}{F}_{\max }$ decreases even in the absence of limb darkening. This is due to the fact that as the impact parameter approaches the edge of the source (β → 0 or b₀ → 1), significant regions of the lens' magnification pattern will lie outside of the source. This effect is more prominent for Event 2, as the Einstein ring of the lens falls farther off the disk of the source. Also note that the wings of the events become much more prominent as b₀ increases, causing f_ws to increase (Equation (18)). This is because μ_rel for these events must decrease, leading to a longer ramp-up and -down. This is fundamentally different than the previous case for the discrepancy in wing shape coming from the ρ − F_S trade-off. For the second model, this change in morphology is actually less significant, as f_ws is less impacted due to the higher value of ρ. Overall, we see that this aspect of the degeneracy must act in concert with matching ${\rm{\Delta }}{F}_{\max }$ for it to become severe.

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Next we investigate the t_* − b₀ degeneracy more quantitatively. Analogous to Figure 7, Figure 9 displays the map of Δχ² values across a grid of (t_*, b₀) pairs of values for the same four values of Γ. We fix all event parameters to those of the fiducial models and find the Δχ² for each (t_*, b₀) pair. Similarly, no minimization is performed here. The analytic prediction of the minimum (Equation (24)) nearly perfectly approximates the calculated minima for these maps. When the limb-darkening profile is steeper, however, ${\rm{\Delta }}{F}_{\max }$ decreases more and more as b₀ increases to chords of the source with surface brightnesses much lower than at the center of the source. The impact of not altering ${\rm{\Delta }}{F}_{\max }$ is evident by the region of small Δχ² becoming more localized near the true values of (t_*, b₀) of the event for increasing Γ. This is clearly shown in the bottom panels, where the traces of the minimum with Δχ² < 9 cover consecutively smaller ranges of b₀ for increasing Γ. In fact, the larger value of ρ in Event 2 results in the range of b₀ values with Δχ² < 9 being smaller than for Event 1 due to the region of magnification falling off of the disk of the source at large impact parameters. In contrast though, the range of t_* values for Event 2 with small Δχ² is wider about the minima. This is because the larger fiducial value of ρ in these events leads to a more commensurate value of f_ws. In the bottom panels, we see that the trace of the minima Δχ² is essentially flat and much smaller than that of b₀ ∈ [0, 0.2] for all values of Γ, as these chord lengths are all essentially the same and the impact of limb darkening is small for these impact parameters.

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In all, this aspect of the degeneracy manifests most strongly in a range of impact parameters that are close to zero, where ${\rm{\Delta }}{F}_{\max }$ is minimally impacted by limb darkening, and chord crossing times vary little. That said, this aspect of the degeneracy is not as significant, as b₀ is not a physically informative parameter, and for much of its range of values, the value of t_* stays relatively constant. If the impact parameter of the event increases too much though, then larger regions of the magnification pattern will lie outside the source, resulting in a deficit in ${\rm{\Delta }}{F}_{\max }$ that must be compensated for by variation of other parameters.

Finally, we consider the t_*–b₀ degeneracy but instead we hold ${\rm{\Delta }}{F}_{\max }$ constant by varying both F_S and ρ. We demonstrate the t_* − b₀ − ρ − F_S degeneracy in Figure 10 for the same six values of Γ. For each Γ, five values of b₀ = 0.0, 0.2, 0.4, 0.6, and 0.8 are chosen. For each, we vary t_* to match t_FWHM and then find the values of f_S and ρ that minimize Δχ² with respect to the fiducial model using least-squares minimization. In all cases in Event 1, we see extremely close agreement between the light curves, except for the b₀ = 0.8 case (the white light curve) where the wings are still more prominent. This is due to the prior that f_S ≤ 3.0, requiring the value of ρ to increase beyond the expected analytic value, causing an increase in f_ws. For Event 2, we see the expected near-perfect degeneracy in the Γ = 0.0 and 1.0 cases, and only slight variations for intermediate values of Γ. The values of the parameters for these light curves are included in the Appendix in Tables 2 and 3 for Events 1 and 2, respectively.

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The quantitative investigation of this four-parameter degeneracy shown in Figure 11 demonstrates its severity. For each fixed pair of (t_*, b₀) values, we vary both f_S and ρ to minimize the Δχ² between the events. To do so, we use a grid search over ρ and a least-squares minimization over f_S. We use a grid search in ρ (rather than least-squares minimization as before) because the minimization algorithm found local minima in unphysical locations far from the trace of the minimum Δχ². Our grid search for ρ was logarithmically spaced over $\mathrm{log}(\rho )=[-2,\mathrm{log}(30)]$, and we maintained the prior that f_S < 3.0. Compared to the plots in Figure 9, we see a significant increase in the area of the Δχ² ≳ 4 values and a dramatic increase in the range of b₀ values out to ∼0.8 that have Δχ² ≲ 1. The extent of this region is slightly larger for Event 2. Although not visible in the lower panel, the average value of Δχ² along the minimum for b₀ < 0.8 dropped by roughly an order of magnitude from Δχ² ∼ 0.1 to ∼ 0.01, and so the degeneracy is more challenging to break via improved photometric precision. By allowing both ρ and f_S to vary for fixed limb darkening, the true nefarious nature of this degeneracy becomes increasingly apparent.

Even when isolating aspects of the degeneracy by varying a subset of parameters, the maximum differences in the light curves can be on the order of 0.1%. The most obvious deviations are in the wings/shoulders of the events with b₀ = 0.8, where the changes in ρ or limb-darkening profile at different impact parameters are strongest. However, we note that these light curves still do not demonstrate the most severe form of the degeneracy discussed in this paper. The most severe form of the degeneracy involves varying t_*, ρ, b₀, F_S, and Γ simultaneously in order to minimize the χ² or the rms of the difference in the light curves with respect to the fiducial light curve. We further explore the isolated degeneracies and the "complete" degeneracy in Section 6.